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Theorem atpointN 35610
Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a  |-  A  =  ( Atoms `  K )
ispoint.p  |-  P  =  ( Points `  K )
Assertion
Ref Expression
atpointN  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )

Proof of Theorem atpointN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . 4  |-  { X }  =  { X }
2 sneq 4042 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
32eqeq2d 2471 . . . . 5  |-  ( x  =  X  ->  ( { X }  =  {
x }  <->  { X }  =  { X } ) )
43rspcev 3210 . . . 4  |-  ( ( X  e.  A  /\  { X }  =  { X } )  ->  E. x  e.  A  { X }  =  { x } )
51, 4mpan2 671 . . 3  |-  ( X  e.  A  ->  E. x  e.  A  { X }  =  { x } )
65adantl 466 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  E. x  e.  A  { X }  =  {
x } )
7 ispoint.a . . . 4  |-  A  =  ( Atoms `  K )
8 ispoint.p . . . 4  |-  P  =  ( Points `  K )
97, 8ispointN 35609 . . 3  |-  ( K  e.  D  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
109adantr 465 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
116, 10mpbird 232 1  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   {csn 4032   ` cfv 5594   Atomscatm 35131   PointscpointsN 35362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-pointsN 35369
This theorem is referenced by: (None)
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